Arithmetic-Geometric Mean Inequality in Excel

The Arithmetic-Geometric Mean Inequality, or AM-GM for short, is a mathematical rule that compares two different ways of finding the average of a list of numbers. The arithmetic mean is what most people think of as the average: you add up all the numbers and divide by how many there are. The geometric mean is another way of finding the average: you multiply all the numbers and take the nth root, where n is how many numbers there are.

The AM-GM inequality says that the arithmetic mean is always greater than or equal to the geometric mean, unless all the numbers are the same, in which case they are equal. For example, if you have two numbers, 2 and 8, their arithmetic mean is (2 + 8) / 2 = 5, and their geometric mean is √(2 × 8) = 4. You can see that 5 is greater than 4, so the AM-GM inequality holds. However, if you have two numbers that are both 4, their arithmetic mean and geometric mean are both 4, so they are equal.

The AM-GM inequality can be used to compare different shapes with the same area or volume, such as rectangles and squares, or spheres and cubes. It can also be used to solve some optimization problems, where you want to find the maximum or minimum value of a function. The AM-GM inequality is one of the most famous and useful inequalities in mathematics, and it has many generalizations and extensions.

Basic Theory:

The AGM inequality states that for any two non-negative numbers $a$ and $b$ , the following inequality
holds:

$\sqrt{ab} \leq \frac{a+b}{2}$

Equality occurs if and only if $a = b$ .

Procedures:

Implementing AGM in Microsoft Excel involves a step-by-step process:

Input Values: Enter the non-negative numbers $a$ and $b$ into designated cells in your
Excel spreadsheet.
Calculate Arithmetic Mean: Use the formula = (a + b) / 2 to find the arithmetic
mean.
Calculate Geometric Mean: Use the formula =SQRT(a * b) to find the geometric
mean.
AGM Inequality Test: Compare the results from steps 2 and 3. If
$\sqrt{ab} \leq \frac{a+b}{2}$ holds true, the AGM inequality is satisfied.

Comprehensive Explanation:

Let’s consider a scenario where $a = 5$ and $b = 8$ . We will follow the procedures outlined above to
demonstrate the AGM inequality.

Real-world Scenario:

	A	B
1	a	5
2	b	8

Calculations:

Arithmetic Mean: $\frac{a+b}{2} = \frac{5+8}{2} = 6.5$

Geometric Mean: $\sqrt{ab} = \sqrt{5 \times 8} = \sqrt{40} \approx 6.32$

AGM Inequality Test:

Since $6.32 \leq 6.5$ , the AGM inequality holds true for the given values of $a$ and $b$ .

Other Approaches:

Manual Calculation: If you prefer not to use Excel functions, you can manually perform the
calculations using a calculator or pen and paper.
Solver Add-in: For more complex scenarios, you can leverage the Solver add-in in Excel to
optimize a function subject to constraints, which can include the AGM inequality.